In mathematics, the Frobenius method describes a way to find an infinite series solution for a second-order ordinary differential equation of the form

We can divide through by z² to obtain a differential equation of the form

which we can solve with regular power series methods if p(z)/z or q(z)/z are analytic at z = 0, but of course these functions are not. The Frobenius method enables us to create a power series solution to such a differential equation.

Explanation

The Frobenius method tells us that we can seek a power series solution of the form

Differentiating:

Substituting:

The expression r(r-1)+p(0)r+q(0)=I(r) is known as the indicial polynomial, which is quadratic in r.

Using this, the general expression of the coefficient of z^k+r is

These coefficients must be zero, since they are to be solutions of the differential equation, so

The series solution with A_k above,

satisfies

If we choose one of the roots to the indicial polynomial for r in U_r(z), we gain a solution to the differential equation. If the difference between the roots is not an integer, we get another, linearly independent solution in the other root.

Example

Let us solve

Divide throughout by z² to give

which has the requisite singularity at z=0.

Use the series solution

Now, substituting

We need to shift the final sum.

We can take one element out of the sums that start with k=0 to obtain the sums starting at the same index.

We obtain one linearly independent solution by solving the indicial polynomial r(r-1)-r+1 = r²-2r+1 =0 which gives a double root of 1. Using this root, we set the coefficient of z^k+r-2 to be zero (for it to be a solution), which gives us the recurrence

Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form.

External links

The Frobenius method can be generalized to orders of ordinary differential equation greater than two, see

• A Generalization of the Frobenius Method for Ordinary Differential Equations with Regular Singular Points (http://www.ansinet.org/fulltext/jms2113-7.pdf) (pdf)